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the_standard_model

The Standard Model

Symmetry

The spacetime symmetry group of the standard model is the Poincare group. The Lie algebra of the standard model is

$$ \mathfrak{su}(3)_C \times \mathfrak{su}(2)_L \times \mathfrak{u}(1)_Y .$$

However, the corresponding Lie group is not necessarily

$$ SU(3)_C \times SU(2)_L \times U(1)_Y \, , $$

but could be

$$ SU(3) \times U(2) $$

or

$$ U(3) \times SU(2) $$

or

$$ S(U(3) \times U(2)) \, , $$

which all have the same Lie algebra. (Source)

There are thirteen connected Lie groups' with the same Lie algebra as $SU(3)\times SU(2) \times U(1)$. […] We can eliminate choices 1—4 by demanding that the gauge group be compact. Choices 9—13 may be removed from consideration by using the fact that the true non Abelian group of the quarks and leptons in the standard model is $SU(3) \times SU(2)$, since color triplet and weak doublet representations exist in nature.Note that the simply connected universal covering group of all 13 groups above is $SU(3) \times SU(2) \times R$, while $SU(3) \times SU(2) \times U(1)$ is the covering group for groups 5—13. As discussed previously by O'Raifeartaigh, there then remain four possible true symmetry groups for the standard model: $$SU(3) \times SU(2) \times U(1)$$ $$ SU(3) \times U(2) $$ $$ U(3) \times SU(2) $$ $$ S(U(3) \times U(2)) \, . $$ [..] It should be understood from the beginning that the groups $U(3) \times SU(2)$, $SU(3) \times U(2)$, and $S(U(3) \times U(2) )$ yield the same perturbative quantum field theory as $SU(3) \times SU(2) \times U(1)$, since perturbative effects depend only on the Lie algebra. Whether or not the four groups lead to different nonperturbative effects is to the author's knowledge still an open question. Global structure of the standard model, anomalies, and charge quantization by Joseph Hucks

What people usually call the gauge group of the Standard Model: SU(3) × SU(2) × U(1) actually has a bit of flab in it: there's a normal subgroup that acts trivially on all known particles. This subgroup is isomorphic to Z/6. If we mod out by this, we get the “true” gauge group of the Standard Model: G = (SU(3) × SU(2) × U(1))/(Z/6) And, this turns out to have a neat description. It's isomorphic to the subgroup of SU(5) consisting of matrices like this: (g 0) (0 h) where g is a 3×3 block and h is a 2×2 block. For obvious reasons, I call this group S(U(3) × U(2)) If you want some intuition for this, think of the 3×3 block as related to the strong force, and the 2×2 block as related to the electroweak force. A 3×3 matrix can mix up the 3 “colors” that quarks come in - red, green, and blue - and that's what the strong force is all about. Similarly, a 2×2 matrix can mix up the 2 “isospins” that quarks and leptons come in - up and down - and that's part of what the electroweak force is about. […] l assume that one way or another, you're happy with the idea of S(U(3) × U(2)) as the true gauge group of the Standard Model! Maybe you understand it, maybe you're just willing to nod your head and accept it.

http://math.ucr.edu/home/baez/week253.html

A great discussion of these things with awesome illustrations can be found in section 1.4 and at page 26 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou:

As a further example, consider the standard electroweak theory. In this case, the gauge group was identified in Section 1.4 as $U(2)$ or $[SU(2)\times U(1)]/Z_2$, by which we meant that $U(2)$ can be obtained from $SU(2) \times U(1)$ by identifying certain pairs of elements as explained in the paragraph after 1.4.9. The group $SU(2) \times U(1)$ itself consists of couples of elements from respectively the groups $SU(2)$ and $U(1)$, which we mai represent symbolically as points inside the rectangle in Figure 2.8 where the vertical axis represents the group $SU(2)$ and the horizontal axis $U(1)$, and the parallel edges of the rectangle are understood to be identified so as to make the rectangle into a “hyper torus”.

page 26 in Some Elementary Gauge Theory Concepts by Hong-Mo Chan, Sheung Tsun Tsou

See also: Gauge Group

Representations

A single SM family (without right-handed neutrino) is the smallest non-trivial chiral anomaly-free representation of $SU(3) \times SU(2) \times U(1)$. (Geng and Marshak (1989))

Fermions = Matter Fields

There are currently $15$ known left-chiral fermions, which also can appear as right-chiral fermions. In addition, all of them, are doublets under the $\mathfrak{su}(2)_L \times \mathfrak{su}(2)_R$ Lorentz symmetry. The left-chiral fermions transform according to the $(2,1)$ representation of the Lorentz algebra, where the numbers denote the dimensions of the representations. The right-chiral fermions transform according to the $(1,2)$ representation. The doublet states describe spin fields with spin up and fields with spin down: $$\left\{(+\frac{1}{2},0),(-\frac{1}{2},0),(0,+\frac{1}{2}),(0,-\frac{1}{2}) \right\} \equiv \left\{(\uparrow,0),(\downarrow,0),(0,\uparrow),(0,\downarrow) \right\}. $$ This means concretely, for example, that we have the following fields that are directly connected to the isospin carrying electron:

  • $e_L^\uparrow$ : the isospin carrying electron, left-chiral with spin up. Representation: $(+\frac{1}{2},0)$.
  • $e_L^\downarrow$ : the isospin carrying electron, left-chiral with spin down. Representation: $(-\frac{1}{2},0)$.
  • $e_L^{C\uparrow}$ : the isospin carrying anti-electron, right-chiral with spin up. Representation: $(0,+\frac{1}{2})$. (Charge conjugation flips chirality.)
  • $e_L^{C\downarrow}$ : the isospin carrying anti-electron, right-chiral with spin down. Representation: $(0,-\frac{1}{2})$.

In addition, we have electron fields which do not carry ispospin:

  • $e_R^\uparrow$ : the isospin zero electron, right-chiral with spin up. Representation: $(0,+\frac{1}{2})$.
  • $e_R^\downarrow$ : the isospin zero electron, right-chiral with spin down. Representation: $(0,-\frac{1}{2})$.
  • $e_R^{C\uparrow}$ : the isospin zero anti-electron, left-chiral with spin up. Representation: $(+\frac{1}{2},0)$.
  • $e_R^{C\downarrow}$ : the isospin zero anti-electron, left-chiral with spin down. Representation: $(-\frac{1}{2},0)$.

Therefore, we have $15 \times 2 \times 2=60$ fermionic degrees of freedom in each generation and $60 \times 3=180$ fermionic degrees of freedom in total.

Various principles of quantum field theory, such as the CPT theorem, require that fermi fields should be real.from Geometry and Physics by E. Witten

Bosons = Gauge Fields

In the standard model there are $12$ gauge boson, with correspond to the $1$ basis generator of $U(1)_Y$, the three basis generators of $SU(2)_L$ and the eight basis generators of $SU(3)_C$. They all transform according to the $(2,2)=(2,0)\otimes (0,2)$ representation of the $\mathfrak{su}(2)_L \times \mathfrak{su}(2)_R$ Lorentz algebra. Therefore, we have for each gauge field three distinct physical configurations:

  • $(+\frac{1}{2},+\frac{1}{2})$, which we call spin up. (Spin $1$ in the direction of measurement.)
  • $(+\frac{1}{2},-\frac{1}{2})\simeq (-\frac{1}{2},+\frac{1}{2})$, where we say the spin is orthogonal to the axis of measurement. (Spin $0$ in the direction of measurement.)
  • $(-\frac{1}{2},-\frac{1}{2})$, which we call spin down. (Spin $-1$ in the direction of measurement.)

Thus, we have in total $12 \times 3=36$ gauge bosonic degrees of freedom in the standard model.

Important Notions

The QCD scale:

The QCD scale can be seen as the scale at which the color gauge coupling becomes sufficiently strong to drive the formation of bound states.

https://arxiv.org/pdf/0910.5167v1.pdf

Problems

Further Reading

the_standard_model.txt · Last modified: 2017/05/03 10:35 by jakobadmin